Two problems on the greatest prime factor of $n^2+1$
Streszczenie
Let $P^+(m)$ denote the greatest prime factor of the positive integer $m$. In [Arch. Math. (Basel) 90 (2008), 239–245] we improved work of Dartyge [Acta Math. Hungar. 72 (1996), 1–34] to show that \[ |\{n \le x: P^+(n^2+1) \lt x^{\alpha } \}| \gg x \] for $\alpha \gt 4/5$. In this note we show how the recent work of de la Bretèche and Drappeau [J. Eur. Math. Soc. 22 (2020), 1577–1624] (which uses the improved bound for the smallest eigenvalue in the Ramanujan–Selberg conjecture given by Kim [J. Amer. Math. Soc. 16 (2003), 139–183]) along with a change of argument can be used to reduce the exponent to $0.567$. We also show how recent work of Merikoski [J. Eur. Math. Soc. 25 (2023), 1253–1284] on large values of $P^+(n^2+1)$ can improve work by Everest and the author [London Math. Soc. Lecture Note Ser. 352, Cambridge Univ. Press, 2008, 142–154] on primitive divisors of the sequence $n^2+1$.
